Abstract

By applying techniques in the theory of convex functions and Schur-geometrically convex functions, the author investigates a conjecture of Satnoianu on an algebraic inequality and generalizes some known results in recent years.

1. Introduction

For and with , let
These quantities are, respectively, called the arithmetic, geometric, and harmonic means of a positive sequence . For more information on the theory of means, see the monograph [1] or the papers [2–5] and plenty of references therein.

This conjecture has been solved and researched in [7–10]. Among them, Wu obtained in [10] the following result.

Theorem 2 (see [10, Theorem 1]). Let be a positive integer and let , , and for be positive numbers. If and
then

It is clear that inequality (5) generalizes (3). Therefore, we would like to call inequality (5) Satnoianu-Wu’s inequality.

Now we naturally pose the following questions: (1)Can one improve the condition (4)? (2)Does the reversed inequality of (5) exist? (3)Are there other types of inequalities of Satnoianu-Wu type?

The goal of this paper is to answer these questions.

2. Definitions and Lemmas

We need the following definitions and lemmas.

Definition 3 (see [11, page 8]). Let and . We say that is majorized by (in symbols ) if
for , where and are rearrangements of and in a descending order.

Definition 4 (see [12]). Let .(1)The set is said to be geometrically convex if for every and .(2)A function is said to be Schur-geometrically convex on if implies for every .(3)A function is said to be Schur-geometrically concave on if implies for every .

Lemma 5 (see [12]). Let be a symmetric and geometrically convex set with inner points and a symmetric and differentiable function in . Then is a Schur-geometrically convex (or Schur-geometrically concave, resp.) function on if and only if

Lemma 6 (see [1, page 4, Bernoulli's inequality]). The inequality
holds for and or for and . If and , inequality (8) is reversed.

3. Main Results

Now we start off to demonstrate our main results.

Theorem 7. Let be defined as in (2) and , let for , and let with . If , then
If , inequality (9) is reversed.

Proof. For and with , since
then(1)for , or for and , the function is convex on ;(2)for , or for and , the function is concave on .By Jensen’s inequality, when , or when and , we have
where . When , or when and , inequality (11) is reversed.Let for , , , and . Then it is easy to show that . Making use of (11), we obtain inequality (9). The proof of Theorem 7 is complete.

Corollary 8. Under the conditions of Theorem 7 and when , if , then
if , inequality (12) is reversed.

Corollary 9. Let , for , and . If , then
if , inequality (13) is reversed.

Proof. This follows from Theorem 7 by considering that if , we have and that if , we have .

Theorem 10. Let , for , with , and defined by (2). If , then
if , inequality (14) is reversed.

Proof. For , let
Then
for . When , or when and , we have
when and , inequality (17) is reversed. Using Lemma 5, we have the following conclusions:(1)if , or if and , the function is Schur-geometrically convex on ;(2)if and , the function is Schur-geometrically concave on .By the fact that
and by Definition 4, if , or if and , we have
if and , inequality (19) is reversed.Letting for leads to . Putting , and in inequality (19) results in inequality (14). The proof of Theorem 10 is complete.

Corollary 11. Under the conditions of Theorem 10 and when , if , then
if , inequality (20) is reversed.

Theorem 13. Let , for , and with . If , then
if , inequality (22) is reversed.

Proof. Since is a convex (or concave, resp.) function on for or (or for , resp.), by Jensen's inequality, if or , we have
if , inequality (23) is reversed.Letting and shows . Further from (23), we obtain inequality (22). The proof of Theorem 13 is complete.

Remark 14. It is clear that inequalities (9) and (22) both generalize inequality (3).

Corollary 15. Let , for , with , and defined as in (2).(1)When , one has
(2)When and , one has
(3)When and , one has
(4)When and , one has

Proof. This follows from utilizing the well-known harmonic-geometric-arithmetic mean inequality
and Theorems 7 to 13.

Corollary 16. Under the conditions of Corollary 15 and when ,(1)if and , one has
(2)if and , one has
(3)if and , one has

Acknowledgments

The author thanks four anonymous referees for their careful corrections to and valuable comments on the original version of this paper. This work was partially supported by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant no. NJZY13159, China.

References

P. S. Bullen, Handbook of Means and Their Inequalities, vol. 560 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.